Following [1]} the Weyl spinor can be expanded in our chosen frame-associated spinor basis
\( \Psi _{ABCD} = \Psi ^{(0)}_{abcd}\ k^{a}_{(A} k^{b}_{B} k^{c}_{C} k^{d}_{D)} + 4\Psi ^{(1)}_{abcd}\ k^{a}_{(A} k^{b}_{B} k^{c}_{C} n^{d}_{D)} + 6 \Psi ^{(2)}_{abcd}\ k^{a}_{(A} k^{b}_{B} n^{c}_{C} n^{d}_{D)} \\+ 4 \Psi ^{(3)}_{abcd}\ k^{a}_{(A} n^{b}_{B}n^{c}_{C} k^{d}_{D)} + \Psi ^{(4)}_{abcd}\ n^{a}_{(A} n^{b}_{B} n^{c}_{C} n^{d}_{D)}.\) 
